The only one piece scope mount I can find for the Marlin 39M is a Weaver #90.

Is there another mount that is preferred?

Assuming I buy and use this mount, do I need to put loc tite on the little screws? The receiver is already drilled and tapped.

The scope I am going to use is a Leupold VX II 1-4X. I spoke to Leupold and they can adjust the parallax for 60 yards, and install a post and crosshair reticle.

What rings should I pick?

You can but those soft screws generally bite well.

The Weaver is about the only light one piece I can think of off the top of my head. You could run Weaver rings, light, tough and will get the scope about as low as anything.

marlin used to ship a dovetail adapter plate with the 39 .I still have the one that came with mine.